The proposed non-stationary POT model uses a time-dependent threshold estimated through quantile regression to model extremes in non-stationary climate data. This approach avoids issues with previous stationary POT models and models that use a fixed threshold. The non-stationary model separates changes in extremes into changes in the threshold and changes in variability. Spatial patterns of increases in high quantiles are influenced by both of these factors. Regression quantiles provide a natural way to set a time-dependent threshold in POT analysis.

1. Estimating extremes in climate model data
by the POT method with non-stationary
threshold
Jan Kyselý, Jan Picek, Romana Beranová

2. Session 2: Nonstationary peaks-over-threshold (POT) method
 11:30-11:50 J. Kyselý (TUL/IAP): Estimating extremes in climate model data
by the POT method with nonstationary threshold
why? (overview, application)
 11:50-12:10 J. Picek (TUL): Statistical aspects of the regression quantiles
methodology in the POT analysis
how? (details of the methodology)
 12:10-12:30 M. Schindler (TUL): How to choose threshold in a POT model?
justifying how (specific setting)

4. Non-stationary extreme value models
Most studies:
non-stationary block maxima (e.g. Kharin and Zwiers, 2005; Laurent and Parey,
2007)
& non-stationary POT models (e.g. Abaurrea et al., 2007; Parey et al., 2007;
Yiou et al., 2006) with invariable (fixed) threshold to delimit extremes
→ the intensity of the Poisson process (i.e. the frequency of upcrossings) is time-
dependent
This work:
different approach based on a time-dependent threshold estimated using
quantile regression (Koenker and Basset, 1978)

5. Block maxima vs. peaks-over-threshold (POT) method
‘block maxima’
(block size = usually 1 year/season)
40
35
TMAX in July, AR(1) simulation [°C]
30
25
20
15
10
0 31 62 93 124 155 186 217 248 279
what is modelled: magnitude of
extremes (GEV)

6. Block maxima vs. peaks-over-threshold (POT) method
‘block maxima’
(block size = usually 1 year/season)
40
35
TMAX in July, AR(1) simulation [°C]
30
25
20
15
10
0 31 62 93 124 155 186 217 248 279
what is modelled: magnitude of
extremes (GEV)

7. Block maxima vs. peaks-over-threshold (POT) method
‘block maxima’ ‘peaks-over-threshold’ (POT)
(block size = usually 1 year/season) (threshold = ‘sufficiently high’ quantile)
×
‘optimum threshold’: maximum
40 40
information is used & events are
‘extreme’ and independent
35 35
TMAX in July, AR(1) simulation [°C]
TMAX in July, AR(1) simulation [°C]
30 30
?
25 25
20 20
15 15
10 10
0 31 62 93 124 155 186 217 248 279 0 31 62 93 124 155 186 217 248 279
what is modelled: magnitude of what is modelled: 1) magnitude of
extremes (GEV) excesses (GPD); 2) frequency of
excesses (Poisson process)

8. Peaks-over-threshold (POT) method in non-stationary data
POT with stationary threshold &
non-homogeneous Poisson process
(intensity depends on time)
40
35
TMAX in July, AR(1) simulation [°C]
30
25
20
15
10
0 31 62 93 124 155 186 217 248 279
(e.g. Abaurrea et al., 2007; Parey
et al., 2007; Yiou et al., 2006)

9. Peaks-over-threshold (POT) method in non-stationary data
POT with stationary threshold & POT with non-stationary threshold
non-homogeneous Poisson process & homogeneous Poisson process
(intensity depends on time) × (threshold depends on time)
40 40
35 35
TMAX in July, AR(1) simulation [°C]
TMAX in July, AR(1) simulation [°C]
30 30
25 25
20 20
15 15
10 10
0 31 62 93 124 155 186 217 248 279 0 31 62 93 124 155 186 217 248 279
(e.g. Abaurrea et al., 2007; Parey
et al., 2007; Yiou et al., 2006)

10. Peaks-over-threshold (POT) method in non-stationary data
when significant trend is present in the data (e.g. warming on the long-term
scale as in climate change simulations) & effective sample size is small
↓
model with a time-dependent threshold and constant intensity
(homogeneous Poisson process) superior to a model with a fixed threshold and
time-dependent intensity (non-homogeneous Poisson process)
40
TMAX in July, AR(1) simulation [°C] 35
30
25
20
15
10
0 31 62 93 124 155 186 217 248 279

11. Peaks-over-threshold (POT) method in non-stationary data
POT with stationary threshold & non- POT with non-stationary threshold &
homogeneous Poisson process homogeneous Poisson process
(intensity depends on time) (threshold depends on time)
×
40 40
35 35 ?
TMAX in July, AR(1) simulation [°C]
TMAX in July, AR(1) simulation [°C]
30 30
25 25
20 20
15 15
10 10
0 31 62 93 124 155 186 217 248 279 0 31 62 93 124 155 186 217 248 279
a constant threshold in a POT model cannot be suitable over longer periods of time: there are
either too few exceedances above the threshold in an earlier part of record (which
enhances the variance of the estimated model), or too many exceedances towards the
end of the examined period (which violates asymptotic properties of the model and leads to
bias), or both the deficiencies are present in the examined samples of ‘extremes’

12. ×
Peaks-over-threshold (POT) method in non-stationary data
POT with stationary threshold & non- POT with non-stationary threshold &
homogeneous Poisson process homogeneous Poisson process
(intensity depends on time) (threshold depends on time)
×
40 40
95%
35 35 regression
quantile
TMAX in July, AR(1) simulation [°C]
TMAX in July, AR(1) simulation [°C]
30 30
25 25
20 20
15 15
10 10
0 31 62 93 124 155 186 217 248 279 0 31 62 93 124 155 186 217 248 279
independence of excesses: declustering
(only maxima of clusters taken)

13. Non-stationary POT method
non-stationary POT model:
• threshold modelled in terms of 95% quadratic regression quantiles
• models estimated over 2001-2100
data: coupled GCMs CM2.0, CM2.1, ECHAM5 over Europe; several SRES emission
scenario simulations over 2001-2100 (A2, A1B, B1, A1FI)
comparison of stationary POT models over selected 30-yr time slices (2021-2050,
2071-2100) with non-stationary POT models
models’ performance evaluated in terms of 20-yr return values of TMAX
(20-yr return value in a non-stationary model defined analogously to the conventional meaning as
a value occurring with a probability 1/20 in a given year)

14. Non-stationary POT method
several models for the Generalized Pareto distribution (GPD) of
exceedances fitted and compared:
Model Scale parameter modeled as Shape parameter modeled as Tested
against
1 log (σ(t)) = σ0 ξ = ξ0 ---
2 log (σ(t)) = σ0 + σ1t ξ = ξ0 1
3 log (σ(t)) = σ0 + σ1t ξ(t) = ξ0 + ξ1t 2
4 log (σ(t)) = σ0 + σ1t + σ2t2 ξ = ξ0 2
5 log (σ(t)) = σ0 + σ1t + σ2t2 ξ(t) = ξ0 + ξ1t 4
pairs of models 1 to 5 compared in terms of likelihood ratio tests
in all examined GCM scenarios, the non-stationary extreme value model selected is
model 2, i.e. model with a linear trend in logarithm of the scale
parameter and constant shape parameter

15. Stationary POT method
Fig. 1: Projected
changes in 20-yr
return values of
TMAX estimated for
30-yr time slices
using the
stationary POT
model in 2071-2100
relative to the
control period
1961-1990.

16. Non-stationary vs. stationary POT method
Fig. 2: Differences
between 20-yr return
values of TMAX
estimated using the
non-stationary POT
model for year 2100
and the stationary
POT model over
2071-2100.

17. Non-stationary POT method
Fig. 3: Differences
between 20-yr return
values of TMAX
estimated using the
non-stationary POT
model for years
2100 and 2071.

18. }
spatial patterns of changes in high
quantiles related to two sources:
1) changes in the location/
threshold (which capture shifts in
the location of the GPD),
} 2) changes in the scale parameter
of the GPD (related e.g. to
interannual variability of extremes)

19. pronounced warming in very high
quantiles of TMAX over western and
central Europe around 45-50°N due to
= shift in the location of the
distribution of extremes
(threshold)
+ BUT maxima in the spatial patterns of
the changes in the 20-yr return values
and the location/threshold do not
correspond exactly to each other, the
former being shifted northward in the
A2, A1B and A1FI scenarios; this is
because of additional changes in
the scale parameter of the GPD,
with a maximum warming around
50-55°N and a cooling south of 45°N

20. SUMMARY 1/2
• The proposed non-stationary POT model with time-dependent threshold
and a homogeneous Poisson process is
 computationally straightforward
 does not violate assumptions of the extreme value analysis (unlike
models with an invariable threshold and a non-homogeneous Poisson process
used in some previous climate change studies, and/or stationary POT models)
• Two sources of increases in high quantiles are disaggregated using the
proposed method: changes in the threshold (95% quantile) & changes in the scale
parameter → climatological interpretation
• Changes in the scale parameter of the distribution of extremes should not be
ignored in climate change studies, as they to a large extent influence spatial patterns
of extremes
• The method may be adjusted to include e.g. circulation indices as other
covariates in addition to time

21. SUMMARY 2/2
Regression quantiles
• a useful concept in mathematical statistics, rarely used in environmental and
climatological studies (mainly for the detection of trends)
• the most natural and intuitive solution to the problem of setting a (time-
dependent) threshold in the POT analysis, corresponding to a high quantile of the
distribution of the examined variable
• the results are not dependent on the particular choice of the threshold: if the 96%
or 97% quantiles are used instead of the 95% quantile, the main findings remain
unchanged